Pro-étale uniformisation of abelian varieties

Abstract

For an abelian variety A over an algebraically closed non-archimedean field K of residue characteristic p, we show that the isomorphism class of the pro-étale perfectoid cover A = lim ← −[p]A is locally constant as A varies p-adically in the moduli space. This gives rise to a pro-étale uniformisation of abelian varieties as diamondsthat works uniformly for all A without any assumptions on the reduction of A.More generally, we determine all morphisms between pro-finite-étale covers of abeloid varieties. For example, over Cp, all abeloids can be uniformised in terms of universal covers that only depend on the isogeny class of the semi-stable reduction over Fp. NotationThroughout, we work over a complete algebraically closed non-archimedean field K of residue characteristic p > 0, with ring of integers O K , maximal ideal m and residue field k. We fix a pseudo-uniformiser ̟ ∈ O K . If char K = 0, we assume that p ∈ ̟O K . We use almost mathematics in the sense of Gabber-Ramero with respect to (O K , m). For any O K -modules M , M ′ we write M a = M ′ if there is a natural almost isomorphism between them. By a rigid space over K, we always mean an adic space of topologically finite type over Spa(K, O K ). By a rigid group over K, we mean a group object in rigid spaces over K, which by convention we always assume to be commutative. Similarly, a perfectoid group over K will mean a commutative group object in the category of perfectoid spaces over K.

Line bundles on rigid spaces in thev-topology

“…is the limit over multiplication by n on X, where n ranges through N. This is represented by a perfectoid space [6, Corollary 5.9] with the interesting feature that it is 'p-adic locally constant in X', that is, many different X have isomorphic 𝑋 [22]. 3.…”

The p-adic Corlette–Simpson correspondence for abeloids

“…Proof Let V be finite free over Z p , then we consider the long exact sequence of Hom(−, V ) applied to the displayed sequence: Since V is derived p-complete and A p is p-divisible, we have by [21,Lemma A.10] Ext v ( A p , V ) = 0.…”

“…This gives an independent proof that a unipotent v-vector bundle is analytic if and only if its associated T p A-representation is trivial on T p C. Remark 4. 21 One way in which this perspective could be helpful is that in contrast to the approach via universal vector extensions, it does not use the group structure on A in an essential way. In particular, we believe that diamantine covers like A c can also help understand when generalised representations are representations beyond the case of abeloids.…”

“…There are plenty of examples of abelian varieties A and B satisfying the assumptions of Proposition 5.18 by the main result of [14]: For example, we can take A and B to be any non-isogeneous simple abelian varieties of good reduction whose special fibres are isogeneous.…”

Perfectoid covers of abelian varieties

“…Remark 5.3. In [10], it is shown that this is a "uniformisation" also in the stronger sense that for two abelian varieties A and A ′ , the perfectoid covers A ∞ and A ′ ∞ are isomorphic whenever A and A ′ are "p-adically close" in some precise sense. In particular, at least locally in the moduli space of abelian varieties we can really think of T p A as a 2d-dimension Z p -sublattice of a fixed perfectoid space determining A, in analogy to the complex uniformisation of abelian varieties.…”